One of the many intriguing observations in Miranda Lund's book Sacred Geometry is that there are only three regular tilings of a plane. That is, triangles, squares, and hexagons are the only three regular polygons that can completely fill a plane if repeated over and over.
There are other ways of filling a plane with combinations of two or more different types of regular polygons, known as semi-regular and demi-regular tilings. But these semi- and demi-regular tilings involve only triangles, squares, hexagons, octagons, or dodecagons: that is, 3, 4, 6, 8, and 12 sided figures. What became of the 5, 7, 9, 10, and 11 sided figures? Assuming that a desperate soul were to try to fill a plane with these figures as regularly as possible, what is the best they could do?
It turns out there are a good many visually interesting arrangements of polygons across a plane that are structurally interesting, without being a regular, demi-regular, or semi-regular tiling. Here's an example that is based on pentagons:
Some of the interesting properties of this arrangement are:
I don't know of a mathematical term for tilings with these types of properties, but I have discovered (or possibly rediscovered) many of them by placing regular polygons point to point. For a time, it became an obsession of sorts to enumerate these various patterns. Polygon Madness is the record of my experiments with such figures.
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© Copyright 2007 by Joseph F. Morales